[seqfan] Re: detective work related to "Floretions"
Jonathan Post
jvospost3 at gmail.com
Sat Nov 21 04:18:59 CET 2009
Yes, perhaps for you, Robert Munafo, but that's what everyone else in
the English speaking world calls it.
Nonassociative Algebra
An algebra which does not satisfy
a(bc)=(ab)c
is called a nonassociative algebra.
SEE ALSO: Algebra, Cayley Number, Complex Number, Division Algebra,
Quaternion, Real Number
REFERENCES:
Kuz'min, E. N. and Shestakov, I. P. "Non-Associative Structures." In
Algebra VI. Combinatorial and Asymptotic Methods of Algebra:
Nonassociative Structures (Ed. A. I. Kostrikin and I. R. Shafarevich).
New York: Springer-Verlag, 1995.
Schafer, R. D. An Introduction to Nonassociative Algebras. New York:
Dover, 1996.
CITE THIS AS:
Weisstein, Eric W. "Nonassociative Algebra." From MathWorld--A Wolfram
Web Resource. http://mathworld.wolfram.com/NonassociativeAlgebra.html
===========
Algebra
The word "algebra" is a distortion of the Arabic title of a treatise
by al-Khwārizmī about algebraic methods. In modern usage, algebra has
several meanings.
One use of the word "algebra" is the abstract study of number systems
and operations within them, including such advanced topics as groups,
rings, invariant theory, and cohomology. This is the meaning
mathematicians associate with the word "algebra." When there is the
possibility of confusion, this field of mathematics is often referred
to as abstract algebra.
The word "algebra" can also refer to the "school algebra" generally
taught in American middle and high schools. This includes the solution
of polynomial equations in one or more variables and basic properties
of functions and graphs. Mathematicians call this subject "elementary
algebra," "high school algebra," "junior high school algebra," or
simply "school algebra," reserving the word "algebra" for the more
advanced aspects of the subject.
Finally, the word is used in a third way, not as a subject area but as
a particular type of algebraic structure. Formally, an algebra is a
vector space V over a field F with a multiplication. The
multiplication must be distributive and, for every f in F and x,y in V
must satisfy
f(xy)=(fx)y=x(fy).
An algebra is sometimes implicitly assumed to be associative or to
possess a multiplicative identity.
Examples of algebras include the algebra of real numbers, vectors and
matrices, tensors, complex numbers, and quaternions. (Note that linear
algebra, which is the study of linear sets of equations and their
transformation properties, is not an algebra in the formal sense of
the word.) Other more exotic algebras that have been investigated and
found to be of interest are usually named after one or more of their
investigators. This practice unfortunately leads to entirely
unenlightening names which are commonly used by algebraists without
further explanation or elaboration.
SEE ALSO: Abstract Algebra, Alternative Algebra, Associative Algebra,
Banach Algebra, Boolean Algebra, Borel Sigma-Algebra, C-*-Algebra,
Cayley Algebra, Clifford Algebra, Commutative Algebra, Derivation
Algebra, Exterior Algebra, Fundamental Theorem of Algebra, Graded
Algebra, Hecke Algebra, Heyting Algebra, Homological Algebra, Hopf
Algebra, Jordan Algebra, Lie Algebra, Linear Algebra, Measure Algebra,
Nonassociative Algebra, Power Associative Algebra, Quaternion, Robbins
Algebra, Schur Algebra, Semisimple Algebra, Sigma-Algebra, Simple
Algebra, Steenrod Algebra, Umbral Algebra, von Neumann Algebra
Portions of this entry contributed by John Renze
On Fri, Nov 20, 2009 at 6:50 PM, Robert Munafo <mrob27 at gmail.com> wrote:
> A term like "nonassociative algebra" might be suitable for some readers, but
> it confuses me. I still think "algebra" is how I solved word problems in 7th
> grade (well not quite, but close :-).
>
> The purpose of my investigations is to make this stuff useable by people who
> need to be shown simple and concrete examples. I understand that
> "commutative" refers to the idea that "3 x 7 = 7 x 3" and that's about it. I
> don't know what "sub-algebra" means, or how "an algebra" can "contain"
> quaternions? If a "projection operator" is something that takes one type of
> number for input and produces another type as output, let's just say that,
> and avoid the jargon. I don't need to have taken college level maths courses
> to to calculate these sequences.
>
> Also, I really do prefer the separate letters a,b,c,d,e,f,...,o,p rather
> than a subscripted coefficient. In my opinion, subscripted coefficients are
> for cases where the number of components is variable, unlimited, and/or
> exceeds the number of letters in your alphabet(s). None of those applies
> here: we have exactly 16 components all the time. It gets worse when there
> are two or more dimensions worth of subscripts, like when defining matrix
> multiplication.
>
> As for the special-named functions like "ibase()", I just need to understand
> and document them, because that's they appear in the existing OEIS database
> entries. Think of my work as a Rosetta stone for Creighton's work (-: There
> were three languages on the Rosetta stone. Here, we have Creighton's
> language, formal algebra language, and "I just want to calculate the d**n
> thing" language.
>
> Jonathan Post wrote:
>> Wasn't there a comment that identified Floretions as a specific
>> nonassociative algebra, in a way that explains where the complex,
>> quaternionic, and octonionic structures come from?
>
> Joerg wrote:
>> Get rid of random letters for the components!
>> As it is, noone will ever dig through this messy notation.
>> [...]
>> Name the components a0, a1, ..., a15.
>> a0 shall be the neutral element (is there such an element?).
>
> Creighton wrote:
>>> Getting rid of the "random letters for the components" is the whole
>>> purpose of introducting the projection operators ibase(X), jbase(X), etc.
> ...
>
> Someone wrote:
>>>> What are the sub-algebras (if it has any)?
>>>> Is it (or are any sub-algebras) associative or commutative?
>>>> Zero-divisors?
>
> Someone wrote:
>> Does your algebra contain (e.g.) complex numbers, quaternions, octonions?
>
> and Someone wrote:
>> Did you make _any_ attempt whatsoever to obtain structural information
> about your algebra?
>
> --
> Robert Munafo -- mrob.com
>
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